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Improved QCD sum rule estimates of the higher twist contributions to polarised and unpolarised nucleon structure functions
Physics Letters B 322 (1994) 425-430 North-Holland
PHYSICS LETTERS B
Improved QCD sum rule estimates of the higher twist contributions to polarised and unpolarised nucleon structure functions Graham G. Ross 1
Department of Theoretical Physics, University of Oxford, Oxford, UK and R.G. Roberts
Rutherford Appleton Laboratory, Chilton, Didcot OXI10QX, UK Received 9 December 1993 Editor: P.V. Landshoff We re-examine the estimates of the higher twist contributions to the integral of gl, the polarised structure function of the nucleon, based on QCD sum rules. By including corrections both to the perturbative contribution and to the low energy contribution we find that the matrix elements of the relevant operators are more stable to variations of the Borel parameter M 2, allowing for a meaningful estimate of the matrix elements. We find that these matrix elements are typically twice as large as previous estimates. However, inserting these new estimates into the recently corrected expressions for the first moments of gl leads to corrections too small to affect the phenomenological analysis. For the unpolarised case the higher twist corrections to the GLS and Bjorken sum rules are substantial and bring the estimate of AQCD from the former into good agreement with that obtained from the Q2 dependence of deep inelastic data.
The recent measurements o f the polarised nucleon structure functions g~,n,d [1-3] raise the question of the consistency of the three measurements Ip,n,d = f~ dxg~ "n'd(X). In particular since the three measurements are at different values of < Q2 >, namely 10.7, 2 and 4.6 GeV 2 for p, n and d, it is crucial to understand the Q2 dependence o f the first m o m e n t o f g~ in order to test the Bjorken sum rule [4] or to extract an estimate o f the nucleon's spin content, Aq, consistent with all three experiments. The higher order corrections to the leading twist expressions have been calculated to O(a 2 ) for the nonsinglet quantities and to O ( a s ) for the singlet contribution but it is the power cqrrections 1/Q 2 from the higher twist operators which have recently [5,6] been shown may play an important role in a consistent analysis of the data. The magnitudes o f the reduced matrix elements o f the relevant higher twist operators
U s, U Ns, V s, V Ns were extracted from a Q C D sum rule calculation by Balitsky, Braun and Kolesnichenko (BBK) [7] and used in the analysis o f ref. [5]. The aim o f the present paper is to sharpen the results o f B B K b y re-examining the computation of<< U s'Ns >>, << V s,Ns >>, including a contribution to the perturbative Q C D side o f the sum rule which was d r o p p e d in the Borel transformation and explicitly retaining the continuum term to the nucleon pole on the low-energy side o f the sum rule. This leads to a significant improvement in the stability of the extracted value of the reduced matrix elements and hence to a more reliable estimate o f the higher twist contribution to the integral o f gl. Moreover the estimated values of the matrix elements are significantly larger than previously but since the coefficient of the twist three piece in the first m o m e n t m o m e n t of g~ has very recently been corrected [8], it turns out that the net higher twist contribution to the m o m e n t is minimal. The improvement in stability and increased magnitude o f the ma-
1 SERC Advanced Fellow. 0370-2693/94/$ 07.00 ~) 1994-Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 3 ) 0 0 0 1 1 - U
425
Volume 322, number 4
PHYSICS LETTERS B
trix elements is also true for the unpolarised case and we find that the correction to the Gross-Llewellyn Smith (GLS) sum rule [9] is sufficient to affect the extracted value of A~-g substantially. Following the procedure of BBK we consider the quantity Fu (p), if we are interested in the operators U s'Ns, given by ~(p)
= i2 1 dx
eivx [ dy (T[rl(x)Uu(y)O(O) ]) (1)
where r/ is the nucleon current. Expanding powers of 1/pZ gives
Fu(p> =pMk75[Ap41n2 ( )_-7 #2 + ~ ( ) ~1
+Bin
) _1_ In (/~2)___~ + C ( _---~
Fu (p)
2 e-m2/M2 e-m2/M2 1 M2 ~< U >> -X
(5)
and the QCD sum rule results from equating eq. (3) to eq. (5) to give
em2/M2] = L-5?UJ
× [2AMafdse-S/M's21n(fl-~-~) (2)
The coefficients A ..... D may be read off from eq. (8) of BBK corresponding to the Q C D evaluation of Fu, including non-perturbative effects due to QCD condensates. The next step is to Borel transform the coefficient of puny 5 in eq. (2) which gives
So
pulkys[2AM2f dse-S/M2s21n(~) 0 + BM4{1 - e-So/~t~} (3)
which differs from eq. (11 ) of BBK since we have explicitly carried out the p2 integration of the ( 1/192) ln(flE/p 2) contribution instead of neglecting it #1 To complete the sum rule the quantity Fu (p) must also be determined in terms of the nucleon and continuum contributions. Balitsky et al. [7 ] use the form .
(4)
~¢1 Balitsky et al. argue that with the natural choice, //~-g 2 _p2 ~ 1 GeV2, this term vanishes. However the Borel transform integrates over p2 so this is, at best, an approximation. Since the effect of the term can be explicitly included we choose not to make this approximation. 426
[22
<< u >> - x M
+D (~4)]
Fu(p ) = -pdk75 ,[[222 (P2 --~ << m2 U>> ~ - + P 2~m2 X ]
where the first term is the pure nucleon pole contribution and the single pole term is added to allow for the interference o f the pole term with a continuum contribution. The Borel transform of the coefficient of PunY5 in eq. (4) is
in
(/~2)__~
-CM2(Ce+ln(I'tz'~}+CMZ+D]\--~--f]
24 February 1994
So + BM4{1 -
e-S~M2} (6)
In this paper we are particularly concerned with estimating the errors in determining the operator matrix elements from the QCD sum rules. Thus we will consider in detail the effects of each of the terms in this expansion and the inclusion of further terms in the Borel expansion in ( M 2)n/n!. On the RHS of eq. (2) the first term not included is ~ 1/p6 which, after Borel transformation contributes a term ,-~ 1/2!M 2 to the RHS of eq. (6). The coefficient of this term should be no bigger than the coefficient of Y if the perturbative expansion in 1/(n! ( M ) " ) is acceptably convergent. By adding such a term and fitting its coefficient we may check this over some range of M 2 and hence establish over what range o f M 2 (if any) the sum rule converges. The corrections to the LHS ofeq. (6) are somewhat more difficult to determine. Further resonance contributions with mass m 2 > m 2 can be included by adding a resonance pole (m2R_p2 ) - 1. After Borel transformation this gives an additional term M 2 e (m2-m2R)/M2Y on the LHS ofeq. (6). However the dominant correction to the LHS ofeq. (2) is not expected to come from a nucleon resonance excitation but from the (N + nn) continuum which has a threshold at E = ~ + m 2, very close to the nucleon pole. As far as we know this contribution has not been considered explicitly even
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PHYSICS LETTERS B
though it is potentially very large. However, we will demonstrate that this contribution does not substantially degrade the accuracy with which we can determine the operator matrix elements provided the resonance term discussed above is a d d e d to eq. (6). The reason is that the n N contribution is well described by the terms proportional to X in eq. (4) together with the resonance term proportional to Y. To demonstrate this we will estimate the contribution due to the N n intermediate state, fig. 2. This gives the term 2
Zf
,
5
1
0~
ilq22 ' < < '
s '
I
. . . .
, 15
'
'
'
'
[
2
'
'
'
' ~
2
InN = gnNN2p m2 x
24 February 1994
5
1
15
2
{ f ( q Z ) 2 [ - 2 ( P + q ) u ~ + (1)75 << U >>] d4q
J
( 2 7 t ) 4 ( q Z - - m 2 ) ( m 2 2 -'~ + q ) 2 ) 2
(7) Here we have included a form factor f (q2) needed to describe the ~/Nrr coupling far from the pion mass shell (on shell we take (~lNn) = ,'].pgnNN). The result is insensitive to the particular choice o f f (q2) provided it provides convergence for large q2; here we choose f ( q 2 ) = qZ/((q2 _ m2)(1 _ q2/0.7 GeV2)) i.e. chosen to vanish as q2 ~ 0 and to be the same as the nucleon electromagnetic form factor for large spacelike q2. (Since f (qZ) is needed for spacelike q2 we have chosen a form factor with no singularities in the spacelike region. ) After taking the Borel transform o f the coefficient of PulkY5 and multiplying by M E e "2/M~/222 we find the resulting contribution to the LHS of eq. (6) is accurately reproduced by the form a + t i M 2 + y M 2 e (m2-m~)/g2
(8)
with a ~ 0.03 << U >>, fl ,-~ 0.15 << U >>, 7 0.2 × << U >>, and m 2 ~ 2m 2. The origin of this form is easy to understand. The zrN singularity starts at (m~ + m ) in the energy plane and, after Borel transformation, gives a contribution proportional to the inte2 -2 . . . . . gral o f e TM -m ) over m, with an appropriate weighting factor. To a good approximation this may be approx. . tra2 m2~lM 2 lmated by the sum of two exponenUals e' - ~" 2 2 2 . and e (m -mR)/M with ml ~ (mn + m ) . Taylor expanding the first term in mn leads to eq. (8). Thus we see that the n N contribution may, to a good approximation, be included in the continuum and interference terms and gives a relatively small correction to the determination of the operator matrix elements.
2
o
~
__
i
s
u2
""
5
t
1
'
'
'
I
15
'
i
,
1
,
i
15
2
,
,
,
b
2
M2 (GeV2) Fig. 1. Values (in GeV 2 ) of the matrix elements << U, V >> derived from fitting the RHS of eq. (6) in bins of 0.2 GeV 2 in M 2 with just the two terms on the LHS (solid lines) and together with the continuum term M 2 e (m2-m2)/M2 Y ( d a s h e d lines).
Using similar methods suggests that the contribution to constant term in eq. (8) from N + nn intermediate states will also be quite small. Let us now turn to the phenomenological analysis and first consider the analysis of ref. [7]. We have four operators to consider U s, U Ns, V s, V Ns the latter two having pdk75 replaced by S~,aAu,vp~puy~ Y5 where A, S stand for symmetric and antisymmetric combination of indices. So we have four Q C D sum rules with coefficients A ~ E given by BBK eqs. (8), (9). In extracting a value o f << U >> from the sum rule, BBK retained only the first two terms on the LHS and also set C = 0. They obtained << U >> at each value o f M 2 by applying ( 1 - M 2 ( d / d M 2 ) ) to the RHS. The stability of this estimate of << U >> relies upon the RHS o f eq. (6) being approximately linear over a range of M 2. The solid lines in fig. 1 are the BBK 427
~7 p
\
r7
', "', ".
"/7
q
Fig. 2. n N contribution, eq. (7), to the correlation function. values for the matrix elements extracted by this pro:edure and the variation with M 2 reflects the importance o f higher derivatives in M 2, making it difficult to arrive at a reliable estimates for << U, V >>. Let us see how the situation changes with the addition o f the proposed continuum term M 2 e(m2-m2)/M2y and the term proposal to C in ~q. (6). As we will demonstrate these are necessary to describe adequately the M 2 behaviour of the RHS aver a reasonable range. We first fit the R H S o f ~q. (6) by the form on the LHS with four parameters << U >>, X, Y and m 2 (the last corresponding to an ~ffective threshold for the continuum) over a wide range 0.5 GeV 2 < M 2 < 2.0 GeV 2. Then, with m 2 fixed (typically around 2 GeV 2), we perform a three parameter fit over each interval of 0.2 G e V 2 in M 2. The dashed line in fig. 2 shows the resulting values af << U >> and we immediately see a marked improvement in the stability c o m p a r e d to neglecting the ;ontinuum term which results in a more meaningful :stimate o f the matrix elements. At the same time we aotice that the actual value obtained is considerably different. At M 2 = 1 GeV 2 for example the new ~stimate is roughly twice the old value. I f instead we :hoose to describe the continuum by e (m2-'n~)/M~ Y (i.e. a double resonance pole) there is a similar improvement in stability and the extracted values o f the matrix elements are within 15% of those in fig. 1. Using these results we will now try to sharpen the ~stimates o f the higher twist contributions further by :onsidering possible additional terms to the R H S of ,=q. (6). We will consider the sensitivity of << U, V >> :o an additional term ~ 1/p6 on the R H S of eq. (2), ..e. a term ,,~ 1 / 2 M 4 on the R H S o f e q . (6). Thus we a~ay have an additional term e m2/m2/2M 2 on the LHS 3f eq. (6) which can be used in fitting the R H S and ~28
24 February 1994
PHYSICS LETTERS B
Volume 322, number 4
help determine the range o f M 2 o v e r which the sum rule is valid. As a result o f fitting the curves (with /~2 = l GeV 2) in fig. 1 with the constraint that the coefficient o f the additional term is no bigger than the coefficient Y of the continuum term lead to an acceptable range 0.8 ~< M 2 ~< 1.75 GeV 2. In fact, the magnitude o f the continuum is correlated, as one might expect, with the value of the hadronic continuum parameter so on the RHS. Our results correspond to so = 2.25 GeV 2 as in refs. [7,10,11]. The dependence on so is weak however and careful fitting reveals that this dependence is absorbed by the explicit continuum contributions, leaving the magnitude o f << U, V >> practically invariant as So varies in the range 1.8 to 5 GeV 2. F o r the above range in M 2, the resulting uncertainties in the values o f << U, V >> from the fitting procedure are comparable to the uncertainties expected from varying/~2 in the range 0.33 to 3 GeV 2. The values o f the reduced matrix elements obtained are << U s >> = 0.046 4- 0.010GeV 2 << U Ms >> = 0.317 4- 0 . 0 1 0 G e V 2 m 2 << V s >> = - 0 . 2 9 2 4- 0.010GeV 2 m 2 << V Ns >> = 0.605 4- 0.030GeV 2
(9)
F r o m the values in eq. (9) we compute the coefficients ap and a , of t h e 1 / Q 2 contributions to the integrals o f gP and g~, using the corrected formulas of ref. [8]
ap+an=
_8~ . 5 [ < <
U s > > _ ¼Yn2 << V s >>]
ap-an = -~.~[<< uNS))-¼m2<<
vNS>>]
(10)
which gives a; = - 0 . 0 2 7 + 0 . 0 0 2 ,
an = - 0 . 0 0 2 4 - 0 . 0 0 2
(11)
The errors in eq. (9) are based simply on the small variation o f the values o f << U, V >> with M 2 and bt2 and are typically 5%. These errors are in addition to the underlying uncertainties arising from the factorisation assumption for the vacuum condensates which are typically 20% [12]. Thus a realistic estimate o f the errors in eq. (11 ) is more like 0.010. The integrals Ip,,~,d can be written
Volume 322, number 4
PHYSICS LETTERS B
24 February 1994
Ip = h + I8 + I o + ap/ Q 2 In = - h
+18 + Io + an/Q 2
Id = 18 + Io + (ap + a n ) / Q 2
(12)
where 13 = ~ 2 [ F + D ]
1-a--Zs-3.58 <<
I8 = 3 ~ [ 3 F - D ]
1 - a__Z_3.58
/t
5
1
.3 - - i
I0 = -~Aq 1 - ~-~
0 s >>
I
I
~
15 I
'
I
I
I
.
.
.
.
(13) <<
Using the measured values of the polarisation asymmetries from refs. [ 1-3 ], the values of Ip,n,d at values o f Q 2 = 10.7, 2, 4.6 GeV 2 were extracted in ref. [6] and determined to be 0.134 ± 0.012, - 0 . 0 2 3 + 0.005, 0.041 ± 0.016 respectively. The estimates for the coefficients ap, an from our improved Q C D sum rule analysis, eq. ( 11 ), when inserted into eqs. (12), (13) yield estimates for the nucleon spin content (for F / D = 0.575 ± 0.016)
0 Ns > >
/ / / / / / / / i .5
1
1.5
e2
(GeV2)
(14)
Fig. 3. Values (in GeV 2 ) of the matrix elements << Os,Ns >> derived from fitting the RHS of eq. (39) of BK [10] with the nucleon double and single pole terms only (solid lines) and together with a continuum term (dashed lines), as in fig. 1.
The value o f Aq obtained from the relatively low Q2 neutron data from SLAC is still out o f line with the values obtained form CERN on the proton and deuterium. If the neutron higher twist coefficient an had come out large and positive, around 0.04 or so, then Aq would decrease a value close to the proton and deuteron estimates. In fact the analysis of Ellis and Karliner [5] used such a value based on the BBK [7] estimates of << U, V >> but with the (now known to be) incorrect formulas for twist three contribution to the first moment. Thus despite the fact that we claim considerably larger estimates for the matrix elements, the corrected formulas [8] lead to a small neutron correction, thereby ruling out higher twists as a way o f reconciling the three experiments. Bag model estimates for an give a zero value [ 13 ]. The i m p r o v e m e n t to our understanding o f the Q C D sum rule estimates of << U, V >> has led to a more meaningful determination o f the higher twist correc-
tions to the integrals of the polarised structure function gL. We recall that two corrections to the sum rule - the evaluation of the ( 1/p2 ) In (fiE/132 ) contribution on the RHS and the inclusion o f the continuum contribution - resulted in more stable estimates of the reduced matrix elements. It is therefore natural to ask if similar corrections apply to other sum rules e.g. the matrix elements << O s'Ns >> which determine the 1/Q 2 corrections to the GLS [9] and Bjorken ( F l ) [ 14] sum rules [10]. The first correction does not apply since ( 1/!02 ) In (#2/p2) terms cancel for the unpolarised operator pulk but the continuum correction should be included. Fig. 3 shows the corresponding determinations o f << O s,Ns >>. In fig. 3a we note the strong variation of << O s >> with M 2, again indicating the inadequacy of the nucleon pole terms alone in eq. (4). Carrying out an analogous fitting procedure as in the polarised case, the resulting estimate for << O s >> is far less
Aq = 0 . 2 4 + 0 . 1 1
fromp
=0.53+0.07
fromn
=0.27±0.15
fromd
429
Volume 322, number 4
PHYSICS LETTERS B
sensitive to the value o f M 2 indicated by the dashed line in fig. 3a. Interestingly, the estimated magnitude o f both matrix elements << O s'Ns >> increases when a realistic continuum term is included. In particular, the estimate used in the analysis of Chyla and Kataev [ 15 ] was that o f ref. [ 10 ] << O s >> = 0.33 ± 0 . 1 6 G e V 2
(15)
which led to a value o f A~-g extracted from the data on xF3 (x, Q2) = 3 GeV 2 on the GLS sum rule o f A (4) = 3 1 8 + 2 3 ( s t a t ) ± 9 9 ( s y s t ) ± 62(twist) MeV MS
(16) If IGLS is the measured value o f the GLS sum rule, then 1
/GLS
3
~s ( Q 2 ) +
-
~r
8 ~ O s >>
2-ff
Q2
(17)
and we see that an increased estimate o f << O s >> leads to a lower value o f A~-g. We estimate the larger and more precise value << O s >> = 0.53 d: 0 . 0 4 G e V 2
(18)
which leads to A (4) = 232 + 23(stat) + 99(syst) + 17(twist) MeV MS
(19) which is more in accord with estimates got from studying the Q2 dependence o f deep inelastic data [ 16], A (4) = 2 3 0 + 55 MeV. Again, the error in eq. (18) MS does not include the intrinsic uncertainty (,,, 20%) associated with the factorisation assumption; including this raises the final error in eq. (19) from 17 to 45 MeV. In summary, we have shown that there are corrections to Q C D sum rules which have not been included in previous determinations o f the relevant reduced matrix elements. These corrections lead to significant i m p r o v e m e n t in the stability o f the extracted values with respect to the range o f the Borel p a r a m e t e r M E. As an example, we have applied these corrections to the case o f the polarised structure function gl o f the nucleon and found that the size o f the the higher twist 430
24 February 1994
contributions is now determined with better precision. We have considered in some detail the corrections to the sum rule. The Nn correction, potentially very significant, has been shown to have little effect on the determination o f the reduced matrix elements. The remaining contributions are under control for a reasonable range of M 2 allowing for a realistic determination o f the error. Our final estimates for the matrix elements, both for polarised and unpolarised structure functions are more reliable and, moreover, significantly larger than previous estimates. Nevertheless because o f the recent corrections to the term multiplying the twist three contribution, the resulting impact on the phenomenology of moments o f g~,,,,d is reduced. For the analysis o f the GLS sum rule however, the resulting value o f fl~s) is more in line with other deep inelastic scattering phenomenology. We are grateful for correspondence with V l a d i m i r Braun.
References
[ 1] EM Collab., J. Ashman et al., Phys. Lett. B 206 (1988) 364; Nucl. Phys. B 328 (1989) 1. [2] SLAC-E142, P.L. Anthony et al., Phys. Rev. Lett. 71 (1993) 959. [3] SM Collab., B. Adeva et al., Phys. Lett. B 302 (1993) 533. [4] J.D. Bjorken, Phys. Rev. 148 (1966) 1467; Phys. Rev. D 1 (1970) 1376. [5] J. Ellis and M. Karliner, Phys. Lett. B 313 (1993) 131; preprint CERN-TH.7022/93 TAUP 2094-93. [6] F.E. Close and R.G. Roberts, Phys. Lett. B 316 (1993) 165. [7] I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Phys. Lett. B 242 (1990) 245. [8] I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Phys. Lett. B 318 (1993) 648(E). [9] D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B 14 (1969) 337. [10] V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. B 283 (1987) 723. [11] B.L. loffe, Proc. XXII Int. Conf. on HEP (Leipzig, 1984) Vol. II, p. 176. [ 12] V.M. Braun, private communication. [13] X. Ji and P. Unrau, preprint MIT-CTP-2232, 1993. [14] J.D. Bjorken, Phys. Rev. 163 (1967) 1767. [15] J. Chyla and A.L. Kataev, Phys. Lett. B 297 (1992) 385. [16] A.D. Martin, W.J. Stirling and R.G. Roberts, Phys. Lett. B 306 (1993) 105.
PHYSICS LETTERS B
Improved QCD sum rule estimates of the higher twist contributions to polarised and unpolarised nucleon structure functions Graham G. Ross 1
Department of Theoretical Physics, University of Oxford, Oxford, UK and R.G. Roberts
Rutherford Appleton Laboratory, Chilton, Didcot OXI10QX, UK Received 9 December 1993 Editor: P.V. Landshoff We re-examine the estimates of the higher twist contributions to the integral of gl, the polarised structure function of the nucleon, based on QCD sum rules. By including corrections both to the perturbative contribution and to the low energy contribution we find that the matrix elements of the relevant operators are more stable to variations of the Borel parameter M 2, allowing for a meaningful estimate of the matrix elements. We find that these matrix elements are typically twice as large as previous estimates. However, inserting these new estimates into the recently corrected expressions for the first moments of gl leads to corrections too small to affect the phenomenological analysis. For the unpolarised case the higher twist corrections to the GLS and Bjorken sum rules are substantial and bring the estimate of AQCD from the former into good agreement with that obtained from the Q2 dependence of deep inelastic data.
The recent measurements o f the polarised nucleon structure functions g~,n,d [1-3] raise the question of the consistency of the three measurements Ip,n,d = f~ dxg~ "n'd(X). In particular since the three measurements are at different values of < Q2 >, namely 10.7, 2 and 4.6 GeV 2 for p, n and d, it is crucial to understand the Q2 dependence o f the first m o m e n t o f g~ in order to test the Bjorken sum rule [4] or to extract an estimate o f the nucleon's spin content, Aq, consistent with all three experiments. The higher order corrections to the leading twist expressions have been calculated to O(a 2 ) for the nonsinglet quantities and to O ( a s ) for the singlet contribution but it is the power cqrrections 1/Q 2 from the higher twist operators which have recently [5,6] been shown may play an important role in a consistent analysis of the data. The magnitudes o f the reduced matrix elements o f the relevant higher twist operators
U s, U Ns, V s, V Ns were extracted from a Q C D sum rule calculation by Balitsky, Braun and Kolesnichenko (BBK) [7] and used in the analysis o f ref. [5]. The aim o f the present paper is to sharpen the results o f B B K b y re-examining the computation of<< U s'Ns >>, << V s,Ns >>, including a contribution to the perturbative Q C D side o f the sum rule which was d r o p p e d in the Borel transformation and explicitly retaining the continuum term to the nucleon pole on the low-energy side o f the sum rule. This leads to a significant improvement in the stability of the extracted value of the reduced matrix elements and hence to a more reliable estimate o f the higher twist contribution to the integral o f gl. Moreover the estimated values of the matrix elements are significantly larger than previously but since the coefficient of the twist three piece in the first m o m e n t m o m e n t of g~ has very recently been corrected [8], it turns out that the net higher twist contribution to the m o m e n t is minimal. The improvement in stability and increased magnitude o f the ma-
1 SERC Advanced Fellow. 0370-2693/94/$ 07.00 ~) 1994-Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 3 ) 0 0 0 1 1 - U
425
Volume 322, number 4
PHYSICS LETTERS B
trix elements is also true for the unpolarised case and we find that the correction to the Gross-Llewellyn Smith (GLS) sum rule [9] is sufficient to affect the extracted value of A~-g substantially. Following the procedure of BBK we consider the quantity Fu (p), if we are interested in the operators U s'Ns, given by ~(p)
= i2 1 dx
eivx [ dy (T[rl(x)Uu(y)O(O) ]) (1)
where r/ is the nucleon current. Expanding powers of 1/pZ gives
Fu(p> =pMk75[Ap41n2 ( )_-7 #2 + ~ ( ) ~1
+Bin
) _1_ In (/~2)___~ + C ( _---~
Fu (p)
2 e-m2/M2 e-m2/M2 1 M2 ~< U >> -X
(5)
and the QCD sum rule results from equating eq. (3) to eq. (5) to give
em2/M2] = L-5?UJ
× [2AMafdse-S/M's21n(fl-~-~) (2)
The coefficients A ..... D may be read off from eq. (8) of BBK corresponding to the Q C D evaluation of Fu, including non-perturbative effects due to QCD condensates. The next step is to Borel transform the coefficient of puny 5 in eq. (2) which gives
So
pulkys[2AM2f dse-S/M2s21n(~) 0 + BM4{1 - e-So/~t~} (3)
which differs from eq. (11 ) of BBK since we have explicitly carried out the p2 integration of the ( 1/192) ln(flE/p 2) contribution instead of neglecting it #1 To complete the sum rule the quantity Fu (p) must also be determined in terms of the nucleon and continuum contributions. Balitsky et al. [7 ] use the form .
(4)
~¢1 Balitsky et al. argue that with the natural choice, //~-g 2 _p2 ~ 1 GeV2, this term vanishes. However the Borel transform integrates over p2 so this is, at best, an approximation. Since the effect of the term can be explicitly included we choose not to make this approximation. 426
[22
<< u >> - x M
+D (~4)]
Fu(p ) = -pdk75 ,[[222 (P2 --~ << m2 U>> ~ - + P 2~m2 X ]
where the first term is the pure nucleon pole contribution and the single pole term is added to allow for the interference o f the pole term with a continuum contribution. The Borel transform of the coefficient of PunY5 in eq. (4) is
in
(/~2)__~
-CM2(Ce+ln(I'tz'~}+CMZ+D]\--~--f]
24 February 1994
So + BM4{1 -
e-S~M2} (6)
In this paper we are particularly concerned with estimating the errors in determining the operator matrix elements from the QCD sum rules. Thus we will consider in detail the effects of each of the terms in this expansion and the inclusion of further terms in the Borel expansion in ( M 2)n/n!. On the RHS of eq. (2) the first term not included is ~ 1/p6 which, after Borel transformation contributes a term ,-~ 1/2!M 2 to the RHS of eq. (6). The coefficient of this term should be no bigger than the coefficient of Y if the perturbative expansion in 1/(n! ( M ) " ) is acceptably convergent. By adding such a term and fitting its coefficient we may check this over some range of M 2 and hence establish over what range o f M 2 (if any) the sum rule converges. The corrections to the LHS ofeq. (6) are somewhat more difficult to determine. Further resonance contributions with mass m 2 > m 2 can be included by adding a resonance pole (m2R_p2 ) - 1. After Borel transformation this gives an additional term M 2 e (m2-m2R)/M2Y on the LHS ofeq. (6). However the dominant correction to the LHS ofeq. (2) is not expected to come from a nucleon resonance excitation but from the (N + nn) continuum which has a threshold at E = ~ + m 2, very close to the nucleon pole. As far as we know this contribution has not been considered explicitly even
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though it is potentially very large. However, we will demonstrate that this contribution does not substantially degrade the accuracy with which we can determine the operator matrix elements provided the resonance term discussed above is a d d e d to eq. (6). The reason is that the n N contribution is well described by the terms proportional to X in eq. (4) together with the resonance term proportional to Y. To demonstrate this we will estimate the contribution due to the N n intermediate state, fig. 2. This gives the term 2
Zf
,
5
1
0~
ilq22 ' < < '
s '
I
. . . .
, 15
'
'
'
'
[
2
'
'
'
' ~
2
InN = gnNN2p m2 x
24 February 1994
5
1
15
2
{ f ( q Z ) 2 [ - 2 ( P + q ) u ~ + (1)75 << U >>] d4q
J
( 2 7 t ) 4 ( q Z - - m 2 ) ( m 2 2 -'~ + q ) 2 ) 2
(7) Here we have included a form factor f (q2) needed to describe the ~/Nrr coupling far from the pion mass shell (on shell we take (~lNn) = ,'].pgnNN). The result is insensitive to the particular choice o f f (q2) provided it provides convergence for large q2; here we choose f ( q 2 ) = qZ/((q2 _ m2)(1 _ q2/0.7 GeV2)) i.e. chosen to vanish as q2 ~ 0 and to be the same as the nucleon electromagnetic form factor for large spacelike q2. (Since f (qZ) is needed for spacelike q2 we have chosen a form factor with no singularities in the spacelike region. ) After taking the Borel transform o f the coefficient of PulkY5 and multiplying by M E e "2/M~/222 we find the resulting contribution to the LHS of eq. (6) is accurately reproduced by the form a + t i M 2 + y M 2 e (m2-m~)/g2
(8)
with a ~ 0.03 << U >>, fl ,-~ 0.15 << U >>, 7 0.2 × << U >>, and m 2 ~ 2m 2. The origin of this form is easy to understand. The zrN singularity starts at (m~ + m ) in the energy plane and, after Borel transformation, gives a contribution proportional to the inte2 -2 . . . . . gral o f e TM -m ) over m, with an appropriate weighting factor. To a good approximation this may be approx. . tra2 m2~lM 2 lmated by the sum of two exponenUals e' - ~" 2 2 2 . and e (m -mR)/M with ml ~ (mn + m ) . Taylor expanding the first term in mn leads to eq. (8). Thus we see that the n N contribution may, to a good approximation, be included in the continuum and interference terms and gives a relatively small correction to the determination of the operator matrix elements.
2
o
~
__
i
s
u2
""
5
t
1
'
'
'
I
15
'
i
,
1
,
i
15
2
,
,
,
b
2
M2 (GeV2) Fig. 1. Values (in GeV 2 ) of the matrix elements << U, V >> derived from fitting the RHS of eq. (6) in bins of 0.2 GeV 2 in M 2 with just the two terms on the LHS (solid lines) and together with the continuum term M 2 e (m2-m2)/M2 Y ( d a s h e d lines).
Using similar methods suggests that the contribution to constant term in eq. (8) from N + nn intermediate states will also be quite small. Let us now turn to the phenomenological analysis and first consider the analysis of ref. [7]. We have four operators to consider U s, U Ns, V s, V Ns the latter two having pdk75 replaced by S~,aAu,vp~puy~ Y5 where A, S stand for symmetric and antisymmetric combination of indices. So we have four Q C D sum rules with coefficients A ~ E given by BBK eqs. (8), (9). In extracting a value o f << U >> from the sum rule, BBK retained only the first two terms on the LHS and also set C = 0. They obtained << U >> at each value o f M 2 by applying ( 1 - M 2 ( d / d M 2 ) ) to the RHS. The stability of this estimate of << U >> relies upon the RHS o f eq. (6) being approximately linear over a range of M 2. The solid lines in fig. 1 are the BBK 427
~7 p
\
r7
', "', ".
"/7
q
Fig. 2. n N contribution, eq. (7), to the correlation function. values for the matrix elements extracted by this pro:edure and the variation with M 2 reflects the importance o f higher derivatives in M 2, making it difficult to arrive at a reliable estimates for << U, V >>. Let us see how the situation changes with the addition o f the proposed continuum term M 2 e(m2-m2)/M2y and the term proposal to C in ~q. (6). As we will demonstrate these are necessary to describe adequately the M 2 behaviour of the RHS aver a reasonable range. We first fit the R H S o f ~q. (6) by the form on the LHS with four parameters << U >>, X, Y and m 2 (the last corresponding to an ~ffective threshold for the continuum) over a wide range 0.5 GeV 2 < M 2 < 2.0 GeV 2. Then, with m 2 fixed (typically around 2 GeV 2), we perform a three parameter fit over each interval of 0.2 G e V 2 in M 2. The dashed line in fig. 2 shows the resulting values af << U >> and we immediately see a marked improvement in the stability c o m p a r e d to neglecting the ;ontinuum term which results in a more meaningful :stimate o f the matrix elements. At the same time we aotice that the actual value obtained is considerably different. At M 2 = 1 GeV 2 for example the new ~stimate is roughly twice the old value. I f instead we :hoose to describe the continuum by e (m2-'n~)/M~ Y (i.e. a double resonance pole) there is a similar improvement in stability and the extracted values o f the matrix elements are within 15% of those in fig. 1. Using these results we will now try to sharpen the ~stimates o f the higher twist contributions further by :onsidering possible additional terms to the R H S of ,=q. (6). We will consider the sensitivity of << U, V >> :o an additional term ~ 1/p6 on the R H S of eq. (2), ..e. a term ,,~ 1 / 2 M 4 on the R H S o f e q . (6). Thus we a~ay have an additional term e m2/m2/2M 2 on the LHS 3f eq. (6) which can be used in fitting the R H S and ~28
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PHYSICS LETTERS B
Volume 322, number 4
help determine the range o f M 2 o v e r which the sum rule is valid. As a result o f fitting the curves (with /~2 = l GeV 2) in fig. 1 with the constraint that the coefficient o f the additional term is no bigger than the coefficient Y of the continuum term lead to an acceptable range 0.8 ~< M 2 ~< 1.75 GeV 2. In fact, the magnitude o f the continuum is correlated, as one might expect, with the value of the hadronic continuum parameter so on the RHS. Our results correspond to so = 2.25 GeV 2 as in refs. [7,10,11]. The dependence on so is weak however and careful fitting reveals that this dependence is absorbed by the explicit continuum contributions, leaving the magnitude o f << U, V >> practically invariant as So varies in the range 1.8 to 5 GeV 2. F o r the above range in M 2, the resulting uncertainties in the values o f << U, V >> from the fitting procedure are comparable to the uncertainties expected from varying/~2 in the range 0.33 to 3 GeV 2. The values o f the reduced matrix elements obtained are << U s >> = 0.046 4- 0.010GeV 2 << U Ms >> = 0.317 4- 0 . 0 1 0 G e V 2 m 2 << V s >> = - 0 . 2 9 2 4- 0.010GeV 2 m 2 << V Ns >> = 0.605 4- 0.030GeV 2
(9)
F r o m the values in eq. (9) we compute the coefficients ap and a , of t h e 1 / Q 2 contributions to the integrals o f gP and g~, using the corrected formulas of ref. [8]
ap+an=
_8~ . 5 [ < <
U s > > _ ¼Yn2 << V s >>]
ap-an = -~.~[<< uNS))-¼m2<<
vNS>>]
(10)
which gives a; = - 0 . 0 2 7 + 0 . 0 0 2 ,
an = - 0 . 0 0 2 4 - 0 . 0 0 2
(11)
The errors in eq. (9) are based simply on the small variation o f the values o f << U, V >> with M 2 and bt2 and are typically 5%. These errors are in addition to the underlying uncertainties arising from the factorisation assumption for the vacuum condensates which are typically 20% [12]. Thus a realistic estimate o f the errors in eq. (11 ) is more like 0.010. The integrals Ip,,~,d can be written
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PHYSICS LETTERS B
24 February 1994
Ip = h + I8 + I o + ap/ Q 2 In = - h
+18 + Io + an/Q 2
Id = 18 + Io + (ap + a n ) / Q 2
(12)
where 13 = ~ 2 [ F + D ]
1-a--Zs-3.58 <<
I8 = 3 ~ [ 3 F - D ]
1 - a__Z_3.58
/t
5
1
.3 - - i
I0 = -~Aq 1 - ~-~
0 s >>
I
I
~
15 I
'
I
I
I
.
.
.
.
(13) <<
Using the measured values of the polarisation asymmetries from refs. [ 1-3 ], the values of Ip,n,d at values o f Q 2 = 10.7, 2, 4.6 GeV 2 were extracted in ref. [6] and determined to be 0.134 ± 0.012, - 0 . 0 2 3 + 0.005, 0.041 ± 0.016 respectively. The estimates for the coefficients ap, an from our improved Q C D sum rule analysis, eq. ( 11 ), when inserted into eqs. (12), (13) yield estimates for the nucleon spin content (for F / D = 0.575 ± 0.016)
0 Ns > >
/ / / / / / / / i .5
1
1.5
e2
(GeV2)
(14)
Fig. 3. Values (in GeV 2 ) of the matrix elements << Os,Ns >> derived from fitting the RHS of eq. (39) of BK [10] with the nucleon double and single pole terms only (solid lines) and together with a continuum term (dashed lines), as in fig. 1.
The value o f Aq obtained from the relatively low Q2 neutron data from SLAC is still out o f line with the values obtained form CERN on the proton and deuterium. If the neutron higher twist coefficient an had come out large and positive, around 0.04 or so, then Aq would decrease a value close to the proton and deuteron estimates. In fact the analysis of Ellis and Karliner [5] used such a value based on the BBK [7] estimates of << U, V >> but with the (now known to be) incorrect formulas for twist three contribution to the first moment. Thus despite the fact that we claim considerably larger estimates for the matrix elements, the corrected formulas [8] lead to a small neutron correction, thereby ruling out higher twists as a way o f reconciling the three experiments. Bag model estimates for an give a zero value [ 13 ]. The i m p r o v e m e n t to our understanding o f the Q C D sum rule estimates of << U, V >> has led to a more meaningful determination o f the higher twist correc-
tions to the integrals of the polarised structure function gL. We recall that two corrections to the sum rule - the evaluation of the ( 1/p2 ) In (fiE/132 ) contribution on the RHS and the inclusion o f the continuum contribution - resulted in more stable estimates of the reduced matrix elements. It is therefore natural to ask if similar corrections apply to other sum rules e.g. the matrix elements << O s'Ns >> which determine the 1/Q 2 corrections to the GLS [9] and Bjorken ( F l ) [ 14] sum rules [10]. The first correction does not apply since ( 1/!02 ) In (#2/p2) terms cancel for the unpolarised operator pulk but the continuum correction should be included. Fig. 3 shows the corresponding determinations o f << O s,Ns >>. In fig. 3a we note the strong variation of << O s >> with M 2, again indicating the inadequacy of the nucleon pole terms alone in eq. (4). Carrying out an analogous fitting procedure as in the polarised case, the resulting estimate for << O s >> is far less
Aq = 0 . 2 4 + 0 . 1 1
fromp
=0.53+0.07
fromn
=0.27±0.15
fromd
429
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PHYSICS LETTERS B
sensitive to the value o f M 2 indicated by the dashed line in fig. 3a. Interestingly, the estimated magnitude o f both matrix elements << O s'Ns >> increases when a realistic continuum term is included. In particular, the estimate used in the analysis of Chyla and Kataev [ 15 ] was that o f ref. [ 10 ] << O s >> = 0.33 ± 0 . 1 6 G e V 2
(15)
which led to a value o f A~-g extracted from the data on xF3 (x, Q2) = 3 GeV 2 on the GLS sum rule o f A (4) = 3 1 8 + 2 3 ( s t a t ) ± 9 9 ( s y s t ) ± 62(twist) MeV MS
(16) If IGLS is the measured value o f the GLS sum rule, then 1
/GLS
3
~s ( Q 2 ) +
-
~r
8 ~ O s >>
2-ff
Q2
(17)
and we see that an increased estimate o f << O s >> leads to a lower value o f A~-g. We estimate the larger and more precise value << O s >> = 0.53 d: 0 . 0 4 G e V 2
(18)
which leads to A (4) = 232 + 23(stat) + 99(syst) + 17(twist) MeV MS
(19) which is more in accord with estimates got from studying the Q2 dependence o f deep inelastic data [ 16], A (4) = 2 3 0 + 55 MeV. Again, the error in eq. (18) MS does not include the intrinsic uncertainty (,,, 20%) associated with the factorisation assumption; including this raises the final error in eq. (19) from 17 to 45 MeV. In summary, we have shown that there are corrections to Q C D sum rules which have not been included in previous determinations o f the relevant reduced matrix elements. These corrections lead to significant i m p r o v e m e n t in the stability o f the extracted values with respect to the range o f the Borel p a r a m e t e r M E. As an example, we have applied these corrections to the case o f the polarised structure function gl o f the nucleon and found that the size o f the the higher twist 430
24 February 1994
contributions is now determined with better precision. We have considered in some detail the corrections to the sum rule. The Nn correction, potentially very significant, has been shown to have little effect on the determination o f the reduced matrix elements. The remaining contributions are under control for a reasonable range of M 2 allowing for a realistic determination o f the error. Our final estimates for the matrix elements, both for polarised and unpolarised structure functions are more reliable and, moreover, significantly larger than previous estimates. Nevertheless because o f the recent corrections to the term multiplying the twist three contribution, the resulting impact on the phenomenology of moments o f g~,,,,d is reduced. For the analysis o f the GLS sum rule however, the resulting value o f fl~s) is more in line with other deep inelastic scattering phenomenology. We are grateful for correspondence with V l a d i m i r Braun.
References
[ 1] EM Collab., J. Ashman et al., Phys. Lett. B 206 (1988) 364; Nucl. Phys. B 328 (1989) 1. [2] SLAC-E142, P.L. Anthony et al., Phys. Rev. Lett. 71 (1993) 959. [3] SM Collab., B. Adeva et al., Phys. Lett. B 302 (1993) 533. [4] J.D. Bjorken, Phys. Rev. 148 (1966) 1467; Phys. Rev. D 1 (1970) 1376. [5] J. Ellis and M. Karliner, Phys. Lett. B 313 (1993) 131; preprint CERN-TH.7022/93 TAUP 2094-93. [6] F.E. Close and R.G. Roberts, Phys. Lett. B 316 (1993) 165. [7] I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Phys. Lett. B 242 (1990) 245. [8] I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Phys. Lett. B 318 (1993) 648(E). [9] D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B 14 (1969) 337. [10] V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. B 283 (1987) 723. [11] B.L. loffe, Proc. XXII Int. Conf. on HEP (Leipzig, 1984) Vol. II, p. 176. [ 12] V.M. Braun, private communication. [13] X. Ji and P. Unrau, preprint MIT-CTP-2232, 1993. [14] J.D. Bjorken, Phys. Rev. 163 (1967) 1767. [15] J. Chyla and A.L. Kataev, Phys. Lett. B 297 (1992) 385. [16] A.D. Martin, W.J. Stirling and R.G. Roberts, Phys. Lett. B 306 (1993) 105.